Download Unit 5 - Middletown Public Schools

Middletown Public Schools
Mathematics Unit Planning Organizer
Grade/Course 10
16 instructional + 4 days for reteaching/enrichment
Duration
Subject
Unit 5
Geometry
Circles and other Conic Sections
Big Idea
Visualize and reason spatially using various attributes of circles.
Apply the relationships among angles and segments related to circles to solve real world measurement problems.
Apply similarity and proportions to determine the relationships between intercepted arcs and circumference; area of sector and area of
circle.
Essential
Question
Make connections between the geometric description and the equation of a circle.
•How can you use theorems about circles to describe relationships among angles, radii, chords and tangents
•What role does the central angle play in helping us find lengths of arcs and areas of regions within the circle?
•How do you describe a circle given its equation? How do you write an equation given a description of a circle?
Mathematical Practices
Practices in bold are to be emphasized in the unit.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Domain and Standards Overview
Grade/Course, Unit Title
2014
1
Date Created/Revised: November 18,
Understand and apply theorems about circles.
Translate between the geometric description and the equation for a conic section.
Priority and Supporting Common Core State Standards
Bold Standards are Priority
CC.9-12.G.C.2 Identify and describe relationships among inscribed
angles, radii, and chords. Include the relationship between central,
inscribed, and circumscribed angles; inscribed angles on a diameter
are right angles; the radius of a circle is perpendicular to the tangent
where the radius intersects the circle.
Explanations and Examples
Examples:
Given the circle below with radius of 10 and chord length of 12, find the
distance from the chord to the center of the circle.
Find the unknown length in the picture below.
CC.9-12.G.C.3 Construct the inscribed and circumscribed circles of a
triangle, and prove properties of angles for a quadrilateral inscribed in a
Grade/Course, Unit Title
2014
Students may use geometric simulation software to make geometric
2
Date Created/Revised: November 18,
circle.
constructions.
CC.9-12.G.C.5 Derive using similarity the fact that the length of the
arc intercepted by an angle is proportional to the radius, and define
the radian measure of the angle as the constant of proportionality;
derive the formula for the area of a sector.
Students can use geometric simulation software to explore angle and
radian measures and derive the formula for the area of a sector.
CC.9-12.G.C.1 Prove that all circles are similar.
Students may use geometric simulation software to model
transformations and demonstrate a sequence of transformations to show
congruence or similarity of figures.
CC.9-12.G.GPE.1 Derive the equation of a circle of given center and
radius using the Pythagorean Theorem; complete the square to find
the center and radius of a circle given by an equation.
Students may use geometric simulation software to explore the
connection between circles and the Pythagorean Theorem.
Examples:
● Write an equation for a circle with a radius of 2 units and center at
(1, 3).
● Write an equation for a circle given that the endpoints of the
diameter are (-2, 7) and (4, -8).
● Find the center and radius of the circle 4x2 + 4y2 - 4x + 2y – 1 = 0.
Students may use geometric simulation software to explore parabolas.
CC.9-12.G.GPE.2 Derive the equation of a parabola given a focus and
Grade/Course, Unit Title
2014
3
Date Created/Revised: November 18,
directrix.
Examples:
● Write and graph an equation for a parabola with focus (2, 3) and
directrix y = 1.
CC.9-12.G.GPE.4 Use coordinate to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure defined by
four given points in the coordinate plane is a rectangle; prove or disprove
that the point (1, √3) lies on the circle centered at the origin and
containing the point (0, 2).
Students may use geometric simulation software to model figures and
prove simple geometric theorems.
Example:
● Use slope and distance formula to verify the polygon formed by
connecting the points (-3, -2), (5, 3), (9, 9), (1, 4) is a
parallelogram.
Concepts
What Students Need to Know
● relationships among
o
o
o
o
o
central angles
inscribed angles
circumscribed angles
radii
chords
Grade/Course, Unit Title
2014
Skills
What Students Need to Be Able to Do
● IDENTIFY
● DESCRIBE
4
Bloom’s Taxonomy Levels
Depth of Knowledge Levels
1
1
Date Created/Revised: November 18,
o tangent
● length of the arc intercepted by an angle
is proportional to the radius
● the radian measure of the angle
● formula for the area of a sector
● DERIVE (using similarity)
5
● DEFINE
1
5
● DERIVE
● equation of circle
● DERIVE
● center and radius of a circle
● FIND (complete the square)
5
3
Learning Progressions
The standards below represent prior knowledge and enrichment opportunities for standards in this unit.
Standard
Prerequisite Skills
Accelerate Learning
CC.9-12.G.C.2 Identify and describe
relationships among inscribed angles, radii,
and chords. Include the relationship
between central, inscribed, and
Understand congruence and similarity using
physical models, transparencies, or geometry
circumscribed angles; inscribed angles on a
software. 8.G.1-5
diameter are right angles; the radius of a
circle is perpendicular to the tangent where
the radius intersects the circle.
CC.9-12.G.C.3 Construct the inscribed and
circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral
Grade/Course, Unit Title
2014
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
5
Date Created/Revised: November 18,
inscribed in a circle.
CC.9-12.G.C.5 Derive using similarity the
fact that the length of the arc intercepted by
an angle is proportional to the radius, and
define the radian measure of the angle as
the constant of proportionality; derive the
formula for the area of a sector.
CC.9-12.G.C.1 Prove that all circles are
similar.
CC.9-12.G.GPE.1 Derive the equation of a
circle of given center and radius using the
Pythagorean Theorem; complete the square
to find the center and radius of a circle
given by an equation.
CC.9-12.G.GPE.2 Derive the equation of a
parabola given a focus and directrix.
CC.9-12.G.GPE.4 Use coordinate to prove
simple geometric theorems algebraically. For
example, prove or disprove that a figure
defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the
point (1, √3) lies on the circle centered at the
origin and containing the point (0, 2).
Grade/Course, Unit Title
2014
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
Understand congruence and similarity using
physical models, transparencies, or geometry
software. 8.G.1-5
6
Date Created/Revised: November 18,
Unit Assessments
Performance Task
Common Formative Assessment
Grade/Course, Unit Title
2014
CC.9-12.G.C.2 Circles
7
Date Created/Revised: November 18,